Questions on the use of algebraic techniques to prove geometric theorems.

learn more… | top users | synonyms (1)

2
votes
3answers
139 views

What is the cone of the conic section?

Given the general (real valued) equation of a conic section: $$ A x^2 + B xy + C y^2 + D x + E y + F = 0 $$ Then what is the circular cone associated with it ? Is it unique ? And is there a way to ...
2
votes
2answers
43 views

Analytic Geometry: One sheeted hyperboloid

Good afternoon! I have a question about analytic geometry. I don't actually know if the answer is quite simple, and I missed something while revising, or if it is actually more complicated than I ...
2
votes
1answer
31 views

Determination of a volume.

Consider, in the Cartesian plane, the square Q having vertices in the points $(-1, 0), (1, 0), (0, -1)$, and $(0, 1)$. The sections of a solid with planes orthogonal to $y=0$ are squares having two ...
1
vote
2answers
42 views

Rotation around a line which is determined by two points in 3D space

If we have three points like $A(x_1,y_1,z_1)$, $B(x_2,y_2,z_2)$ and $C(a,b,c)$. Then, $A$ and $B$ determines a line like $l$. After that, we rotate $C$ around $l$ by $\omega$ degree (anti-clockwise). ...
0
votes
0answers
60 views

A book on analytic geometry

It's easy to find good recommendation for books here for any subject other than analytic geometry ,therefore I'd like to ask for any suggestion of analytic geometry books ,the only charactrestic I'm ...
2
votes
0answers
65 views

A variational strategy for finding a family of curves?

In a recent question, I asked for examples of families of distinct smooth curves with fixed area and perimeter (which for this question I will dub as doubly-isometric). That wording allows $C^1$ ...
0
votes
3answers
74 views

Equation of rectangle

I need equation of a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? for example equation of ellipse is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
3
votes
2answers
252 views

Proof of Descartes' theorem

I came across the use of Descartes' theorem while solving a question.I searched it but I could only find the theorem but not any ...
0
votes
1answer
48 views

Given that the graph of $f$ passes through the point $(1, 6)$ and that the slope of its tangent line at $(x, f(x))$ is $2x + 1$, find $f(2)$.

As in the title - we assume that the graph of $f$ passes through $(1,6)$ (i.e. $f(1) = 6$) and that the slope of its tangent line at $(x, f(x))$ is $2x + 1$ and we are asked to find $f(2)$. How does ...
1
vote
1answer
31 views

What is the basic idea of homogenisation of an equation?

I do get that when you are homogenising it makes it in an equation of pair of straight lines passing through origin but what is its actual point and its applications?
1
vote
1answer
50 views

Lines joining origin to points of intersection of two conics

If the lines joining origin and point of intersection of curves $$ax^2+2hxy+by^2+2gx=0$$ and $$a_1x^2+2h_1xy+b_1y^2+2g_1x=0$$ are mutually perpendicular, then prove that $$g(a_1+b_1)=g_1(a+b)$$ How ...
0
votes
1answer
56 views

Coordinate Geometry of circles; Radical Axis question

If one of the diameters of the circle $x^2+y^2-2x-6y+6=0$ is a chord to the circle with center at $(2, 1)$, then the radius of the second circle is? Apparently the solution is $3$, with the ...
2
votes
1answer
55 views

Help with simple rotation on an x,y plane

I'm a programmer, with too little background in mathematics, and I am currently faced with the challenge of rotating an object on a 2 axis plane. Something that is hopefully quite easy for you guys. ...
2
votes
2answers
38 views

Calculus III: Find the points of the curve…

I have to find the points of the curve $$r\left( t \right) =\left( t,{ t }^{ 2 },{ t }^{ 3 } \right) $$ where the osculating plane passes through the point $\left( 2,-\frac { 1 }{ 3 } ,-6 \right)$.
0
votes
1answer
20 views

Coordinates rotation and function change

In the Cartesian coordinates $(x,y)$, I have a vector function $\bar{f}(x)=\hat{x}A\cos(yk)$, where $A$ and $k$ are constants. I make now a 45 degrees rotation (in the same plane) to the new set of ...
0
votes
1answer
34 views

Three planar vectors $x,y,z$ such that $x$ is orthogonal to $y + z$ and $z$

Let $x$ be a non-zero vector, orthogonal to vectors $y + z$ and $z$, with $x, y, z \in \mathbb R^2$. Prove that $y$, $y - z$ and $z - y$ are orthogonal to $x$ and parallel to $z$. To prove they ...
1
vote
0answers
67 views

intersection of a line and plane on a 3-sphere

Suppose I have two 4D points, $\mathbf{a}=(a_1,a_2,a_3,a_4)$ and $\mathbf{b}=(b_1,b_2,b_3,b_4)$, that both lie on a unit 3-sphere (i.e. unit distance from origin). In addition, I have a 2-D plane that ...
9
votes
7answers
463 views

Constructing a family of distinct curves with identical area and perimeter

Two recent questions were posed by Arjuba [1] [2] asking for counter-examples regarding whether two different figures could have the same perimeter and area. Responders quickly raised a number of such ...
1
vote
4answers
56 views

Doubts on locus and its equation

"Find the equation to the locus of a point which is collinear with points $M(a,0)$ and $N(0,b)$." The answer is $- x/a + y/b$. How I tried to find the solution: $P$ is a point whose assigned ...
0
votes
2answers
53 views

Determine the matrix of the reflections in the fol­lowing plane in $\Bbb R^3$. [closed]

$2x_1-2x_2-x_3= 0$ How would I go about approaching this problem?
2
votes
2answers
106 views

Parametric formula for figure 8 mobius strip

I'm making 3D prints with Mathematica, and am interested in a parametric formula for a mobius strip that is in the form of a figure 8, rather than simply a circle with a twist in it. Can someone help ...
0
votes
1answer
44 views

Equation for the length of a chord parallel to either the minor or major axis in an ellipse

I am looking for a way to compute the length of any chord parallel to the minor (or major) axis of an ellipse. In all cases I know the lengths of both axes, and the distance between the chord and axis ...
0
votes
0answers
7 views

Find gap coordinates to connectable reference Rectangle

Sorry for my english. This is my first question. I need to find gap coordinates of items to reference object programmaticly. Item positions can be changed. How can i do this with an efficent way? ...
1
vote
1answer
29 views

Line not intersecting circle, maximum value of expression involving radius

If line $y+x=2$ do not intersect any member of circles $x^2 + y^2 -ax = 0$ at two distinct points where a is parameter, then maximum value of $|a + 4|$. My try: Since the line does not intersect ...
-1
votes
3answers
454 views

getting the slopes of the sides of an equilateral triangle given 2 points

I want to get the slopes of an equilateral triangle given the 2 vertices. Let's say they are (0, 0) and (5, 5). Graphing this would give 2 triangles forming a diamond. I tried to use distance formula ...
1
vote
2answers
47 views

How to parameterize the maximum of a function?

$f(x) := -4(\frac{1}{2}-x)^2+1$ Is it possible to construct a parameterized version of $f$, say $f_a$, which fulfills: $f_a(0) = f_a(1) = 0$ $f_a(a) = 1$ $f_a'(x) = 0 \Leftrightarrow x=a$ $f_a''(a) ...
1
vote
3answers
94 views

Curiosity about Kronecker's Delta?

My professor gave this subject in the class (analytic geometry) and I thought it was very complicated, then I just decided to open Wikipedia entry on Kronecker's Delta and discovered it is quite ...
1
vote
2answers
104 views

Omitting $i$ in calculations

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian ...
5
votes
3answers
222 views

Calculate the distance from a point to a line

Por favor, alguém me ajude com essa questão de Geometria: Please, can someone help me with this geometry question? Given the point $A(3,4,-2)$ and the line $$r:\left\{\begin{array}{l} x = 1 + t ...
0
votes
1answer
65 views

Central angle of an ellipse

If I have an ellipse centered at the origin and know the length of $a$ and $b$ and was given the length of an arc, how can I find the angle that is between the two radius from the center of the ...
-1
votes
1answer
22 views

Preserving incidence relation proof

How can one prove via analytic method that projective map preserves incidence relation?
1
vote
0answers
32 views

normalized mean curvature flow with convex initial hypersurface has finite velocity

I can't understand the prove in [Xi-Ping Zhu] Lectures on mean curvature flows. The statement as follow. Lemma 3.5 (page 32) There exists a positive constant $C$ such that ...
0
votes
1answer
28 views

Line parallel to plane

See if the line e is parallel with to the plane $α$. If not, find the intersecting point. $$\begin{align} α: & \quad \quad x-3y+z+1=0 \\ e: & \quad \{x+y-z=3, 2x-y-4z=3\} \end{align}$$
1
vote
1answer
164 views

Equation of parabola, tangent at vertex [closed]

Two tangents on a parabola are $x-y=0$ and $x+y=0$. If $(2,3)$ is the focus of the parabola, then find the equation of tangent at the vertex. Thanks. My thoughts: Can't figure out anything :(
0
votes
1answer
22 views

Determine Center Point based on 2 separate elipses

First timer here. I've been digging back into my good old maths days but am extremely rusty (beyond belief). I got a really tricky question that i want to determine formula for so that my mate can ...
-1
votes
4answers
57 views

Find the line through $(-1,4)$ for which the distance to $(6,3)$ is 5

This is the question: Find the line through $(-1,4)$ for which the distance to $(6,3)$ is $5$ The answer is: $y-4=-4/3(x+1)$ and $y-4=3/4(x+1)$ I do not know how to get this answer. ...
2
votes
1answer
31 views

Definition of (hyper)planes

I know the definition of a plane to be: $(r-r_0)\cdot n = 0$ where $n$ is the vector perpendicular to the plane, $r$ the vector to a given point and $r_0$ the vectors to the points which constitute ...
1
vote
4answers
32 views

Showing that a circle is “tangent” to the $x$-axis if and only if $\left|k\right| = r$.

The problem is this: to show that a circle of radius $r$ and center $(h, k)$ intersects the $x$-axis at exactly one point if and only if $\left|k\right| = r$. Using geometrical intuition, this ...
9
votes
1answer
110 views

Intuition for a certain tensor product.

Tensor products occur in lots of places and until recently I thought I understood them at least reasonably well. During the past few weeks, however, I've attended several talks where the tensor ...
7
votes
3answers
580 views

Can asymptotes be curved?

When I was first introduced to the idea of an asymptote, I was taught about horizontal asymptotes (of form $y=a$) and vertical ones ( of form $x=b$). I was then shown oblique asymptotes-- slanted ...
1
vote
1answer
83 views

How can I convert the following parametric equation to cartesian equation?

\begin{align} x&=\left(1 + \frac{1}{\,\sqrt{\,2t^{2} - 4t + 4\,}\,}\right)t\ -\ 2 \\[3mm] y&=\left(1 - \frac{1}{\,\sqrt{\,2t^{2} - 4t + 4\,}\,}\right)t\ +\ \frac{2}{\,\sqrt{\,2t^{2} - 4t + ...
1
vote
0answers
23 views

How can I find the volume of this prism and points B, C, D and F?

In the triangular prism, A = (0, -1, 1), E = (0, -3, 0). C and D belong to line s: x - 1 = y = y - z. How can I find the prism's volume and the coordinates of B, C, D and F points?
2
votes
3answers
55 views

Find a specific vector equation of a line that divides a angle in half.

I've been studying a little geometry on my own, and I just recently stumbled on this problem, that I'm unable to answer: Given the points A=(2,-1), B=(5,4) and C=(-7,8), find a vector equation of a ...
1
vote
1answer
39 views

Doubt on locus of a median point

I'm learning about geometric locus and ain't had an good time, I'm struggling with this problem: By the way, any study resource on geometric locus is welcome! Given an segment $AB$ formed by points ...
1
vote
1answer
34 views

finding a point of intersection

I need to find a point on the $y-axis$ so the tangents from that point to circles: $(x-6)^2+(y-3)^2=16$, $(x-4)^2+(y-6)^2=5$ are equal in length. I tried to use $(x-a)(x_1-a)+(y-b)(y_1-b)$ but it ...
0
votes
1answer
23 views

The number of points in the rectangle which lie on the curve $y^2=x+\sin x$ and at which the tangent to the curve is parallel to the $X-$axis

The number of points in the rectangle : $\{(x,y)|-10\le x\le10$ and $-3\le y\le3\}$ which lie on the curve $y^2=x+\sin x$ and at which the tangent to the curve is parallel to the $X-$axis, is A) $0$ ...
0
votes
3answers
29 views

Analytic geometry - Mutual tangent for circle and ellipse

The problem I'm trying to solve is : Given a circle of equation $x^2+y^2=4$ ,an ellipse of equation $2x^2+5y^2=10$ and their mutual tangent whose equation is $y=kx+n$, determine $k^2+n^2$. I would ...
1
vote
1answer
54 views

Equation of pair of reflected straight lines given the equation of pair of incident straight lines

If $ax^2 + 2bxy + by^2 = 0$ represents a pair of lines, then find the combined equation of lines that can be obtained by reflecting these lines about the x-axis. I know that this can be done by ...
1
vote
2answers
66 views

Locus of vertex of triangle moving inside circle

A right triangle with sides $3,4$ and $5$ lies inside the circle $2x^2+2y^2=25$. The triangle is moved inside the circle in such a way that its hypotenuse always forms a chord of the circle. The locus ...
0
votes
1answer
38 views

What is the approach required for questions in which you least expect that the graphs meet?

Find the no. of solutions of x in these two equations: (A)$2^x=x^2+1$ (B)$e^x=2x^2$ Both are of the same type, that is, the answer is the least you can expect. (When you plot it on a grapher, you ...